Fig 1.
Schematic illustration of immunoprotective encapsulation device containing human embryonic stem cell-derived β clusters (hES-βC).
Fig 2.
An example of a domain Ω = ΩF ∪ ΩP demonstrating the two inlets at the top and the two outlets
at the bottom in 2D (left) and 3D (right).
Fig 3.
Graphical representation of the diffuse interface approach in one dimension.
Top: The phase-field function Φ. Bottom: The gradient of Φ used to approximate the location of the interface.
Fig 4.
The phase-field function for three different network configurations considered in this work.
The zoom-in inserts show the computational mesh. The mesh is refined in the areas with large |∇Φ|, which approximate the interface.
Fig 5.
A steady-state solution for the Stokes and Darcy pressure (left) and velocity magnitude (middle) obtained using the sharp interface model and the diffuse interface model.
The right two panels shows concentration obtained at Tt = 200. A plot of each of these variables over the line indicated in the leftmost panel is shown in Fig 6.
Fig 6.
The left three panels show the pressure, velocity and concentration, respectively, obtained using the sharp and the diffuse interface method, plotted over the line indicated in the leftmost panel in Fig 5.
The right three panels show errors for the pressure, velocity and concentration, respectively, obtained at steady state.
Fig 7.
Total velocity magnitude (left) and concentration (right) in a network consisting of straight channels (top), bifurcating channels (middle) and a hexagonal geometry (bottom).
Fig 8.
Pressure (left) and the Darcy velocity (right) in a network consisting of straight channels (top), bifurcating channels (middle) and a hexagonal geometry (bottom).
Fig 9.
Geometries consisting of narrow straight channels (left) and narrow zigzag channels (right).
Fig 10.
Total velocity magnitude (left) and concentration (right) in a network consisting of narrow straight channels (top) and narrow zigzag channels (bottom).
Fig 11.
Pressure (left) and the Darcy velocity (right) in a network consisting of narrow straight channels (top) and narrow zigzag channels (bottom).
Fig 12.
Function χopt superimposed with the phase-field function indicating the channel geometry.
Regions in red show areas where the insulin production is inhabited, while regions in blue show areas of uninhibited insulin production.
Table 1.
The area where the concentration is larger than the oxygen threshold of uninhibited maximal insulin production for all the geometries considered in this manuscript.
Fig 13.
Computational Domain: The computational domain was discretized using conformal tetrahedral elements, resulting in a total of 926K elements.
Fig 14.
Left: 2D simulation showing free fluid and Darcy velocity ranging from 0 to 13 cm/s. Middle: 3D simulation showing velocity in the inlet, outlet and hexagonal geometry ranging from 0 to 13 cm/s. Right: 3D simulation showing only Darcy velocity ranging from 0 to 4.8 cm/s.
Fig 15.
Left: 2D simulation showing free fluid and Darcy pressure. Middle: 3D simulation showing Darcy pressure. Right: 3D simulation showing pressure in the inlet, outlet and hexagonal geometry. The pressure scale is the same in all three panels—from 0 to around 6,000 dyn/cm2.
Fig 16.
Streamlines generated by the fluid and Darcy velocity.
The color of the streamlines corresponds to the velocity magnitude. Left: 3D simulations, side view. Middle: 3D simulations, front view. Right: 2D simulations.
Fig 17.
Displacement at the steady state.
Top left: Magnitude of displacement obtained in 3D. Top right: Displacement vector field obtained in 3D. Bottom: The displacement obtained in 2D using a model with the spring term γEη (left) and by taking γE = 0 and fixing the center of each poroelastic region (right).
Fig 18.
A series of screenshots depicting the transport of oxygen concentration from t = 0 s to t = 1 s, when the steady state solution is reached.